Sharp bounds on $p$-norms for sums of independent uniform random variables, $0 < p < 1$

TL;DR

This paper establishes precise lower bounds on the $p$-norms of sums of independent uniform variables and explores similar bounds for $p$-Rényi entropy, advancing understanding of these measures for $0 < p < 1$.

Contribution

It introduces sharp bounds on $p$-norms and $p$-Rényi entropy for sums of independent uniforms, filling gaps in the case where $0 < p < 1$.

Findings

Derived sharp lower bounds on $p$-norms in terms of variance.

Extended bounds to $p$-Rényi entropy for the same range.

Provides theoretical tools for analyzing sums of uniform variables.

Abstract

We provide a sharp lower bound on the $p$-norm of a sum of independent uniform random variables in terms of its variance when $0 < p < 1$. We address an analogous question for $p$-R\'enyi entropy for $p$ in the same range.